Symmetry Groups and the Observability of PDEs

Symmetry groups of PDEs allow to transform solutions continuously into other solutions. In this contribution, we use symmetry groups for studying the observability of systems of nonlinear PDEs with input and output. Based on a differential‐geometric representation of the system, we present conditions for the existence of special symmetry groups that do not change the trajectories of the input and the output. If such a symmetry group exists, the system cannot be observable.


Introduction
In this paper, we consider nonlinear PDEs of the form ∂ t x α (z, t) = f α (z, t, x(z, t), ∂ z x(z, t), ∂ 2 z x(z, t), u(t)) , α = 1, . . . , n x (1) on a 1-dimensional spatial domain Ω = (0, 1) ⊂ R with a single input u(t), boundary conditions g λ (t, x(0, t), ∂ z x(0, t)) = 0 , λ = 1, . . . , n A h µ (t, x(1, t), ∂ z x(1, t)) = 0 , µ = 1, . . . , n B , and an output function defined at some point z 0 ∈Ω. ByΩ = [0, 1] we denote the closure of Ω. The functions f α , g λ , h µ , and c are smooth, and we assume that solutions of the PDEs (1) with the boundary conditions (2) exist and are uniquely determined by the initial condition x(z, 0) and the input function u(t) (well-posedness, see e.g. [1]). A pair of initial conditions of such a system is said to be indistinguishable, if for every admissible trajectory u(t) of the input the system generates for both initial conditions the same trajectory y(t) of the output. The system is said to be observable, if (locally) there exists no pair of indistinguishable initial conditions (see e.g. [2]). In this contribution, we want to use the existence of certain symmetry groups for proving that a system is not observable.
In [3], symmetry groups have already been used to deal with a special case of the observability problem with a fixed choice of the input trajectory u(t). Furthermore, in [4] they have also been used to study the accessibility of a system.

Geometric Framework
The mathematical framework for the calculation of symmetry groups is differential geometry. For a differential-geometric representation of the PDEs (1), we introduce the bundle , and π is the canonical projection given in coordinates by π : (z, t, x, u) → (z, t). The first jet manifold J 1 (E) has coordinates (z, t, x, u, x z , x t , u z , u t ), i.e. the coordinates of E plus the derivatives of x and u with respect to z and t. Accordingly, the second jet manifold J 2 (E) has coordinates (z, t, x, u, x z , x t , u z , u t , x zz , x zt , x tt , u zz , u zt , u tt ), with derivatives up to order two. In this framework, the PDEs (1) can be interpreted as algebraic equations which describe a submanifold S 2 ⊂ J 2 (E). Likewise, the boundary conditions i.e. with all coordinates of J 1 (E) except for z.

Symmetry Groups and Non-Observability
Roughly speaking, a symmetry group of a system of PDEs is a (local) group of transformations acting on the manifold E, with the property that it transforms solutions of the system into other solutions, see [5]. A transformation group on E is generated by a smooth vector field v on E, and the conditions, which a transformation group must satisfy to qualify as a symmetry group of the system of PDEs, can be formulated in terms of this vector field. These conditions involve the prolongations j 1 (v) and j 2 (v) of v to the first and the second jet manifold. In the following theorem, we have supplemented the conditions that can 2 of 2 Section 20: Dynamics and control be found in [5] by additional conditions that ensure that the symmetry group also respects the boundary conditions, and that it does not change the trajectories u(t) and y(t) of the input and the output. If such a special symmetry group exists, the system (1) -(3) cannot be observable.

Theorem 3.1 Consider the system (1) with boundary conditions (2) and output (3). If there exists a smooth vector field
. . , n x and v| t=0 = 0 that satisfies the conditions on the submanifold S 2 ⊂ J 2 (E), the conditions on the submanifolds S 1 A ⊂ B A and S 1 B ⊂ B B , and the condition then the system is not observable.
The conditions (4) and (5) guarantee that the vector field v generates a symmetry group of the system with boundary conditions. Geometrically, condition (4) means that the vector field j 2 (v) is tangent to the submanifold S 2 ⊂ J 2 (E) determined by the system equations, and condition (5) means that the vector fields j 1 (v) z=0 and j 1 (v) z=1 are tangent to the submanifolds S 1 A ⊂ B A and S 1 B ⊂ B B determined by the boundary conditions. Since the vector field v has no component in ∂ u -direction and because of condition (6), the symmetry group transforms every solution into other solutions without changing the trajectories of input and output. The initial condition of a transformed solution is indistinguishable from the initial condition of the original solution (in the sense of Section 1). Thus, for every initial condition the symmetry group allows to construct a set of indistinguishable initial conditions, and we can conclude that the system is certainly not observable.
It is worth mentioning that the obtained conditions (4), (5), and (6) are linear PDEs in the unknown coefficients v α x of the vector field v, even though the original system is nonlinear.

Linear Systems
of the system (1) - (3), it turns out that the coefficients v α x (z, t) of the vector field v must be solutions of the homogenous part of the PDEs (7) that satisfy the boundary conditions (8) and produce an output (9) which is identically zero. Since it is well-known that a linear system allows such solutions if and only if it is not observable (see e.g. [6]), in the linear case the conditions of Theorem 3.1 become necessary and sufficient for non-observability. Furthermore, the symmetry group which is generated by v transforms solutions exactly in such a way that the difference of the initial conditions is an element of the non-observable subspace.